Sensitivity and constraints to the 2HDM soft-breaking $Z_2$ parameter $m_{12}$

In this letter we study the specific sensitivity and constraints to the soft breaking $Z_2$ parameter $m_{12}$ of the Two Higgs Doublet Model (2HDM). $m_{12}$ is considered here as an input parameter of the model together with the masses of the Higgs bosons, $m_h$, $m_H$, $m_A$, $m_{H^\pm}$, the ratio of the two Higgs vacuum expectation values, $\tan \beta$, and $\cos(\beta-\alpha)$, with $\alpha$ and $\beta$ the mixing angles of the $\mathcal{CP}$-even and $\mathcal{CP}$-odd Higgs sector, respectively. We explore in particular the sensitivity to $m_{12}$ in the decays of the lightest Higgs boson $h$ to two photons, assuming: 1) that $h$ is the state observed at the LHC at $\sim 125 \ \mathrm{GeV}$, and 2) the $h$ couplings to the SM particles are SM-like, by going to the alignment limit, $\cos(\beta-\alpha) = 0$. The aim of this work is to demonstrate possible constraints on $m_{12}$ from the present measurement of the di-photon signal strength, $\mu_{\gamma \gamma}$. These constraints are relevant in the region of the 2HDM parameter space that is allowed by all the other constraints, specifically, theoretical constraints as well as experimental constraints from LHC Higgs rate measurements, Higgs boson searches, flavor physics and precision observables. The exploration is performed in the four different Yukawa types of 2HDM, where the allowed region by all the other constraints depends on the model type. We also analyze the case that the Higgs-boson mass spectrum accommodates a possible new world average of $m_W$ including the recent CDF measurement.